With the fast development of mobile edge computing (MEC), there is an increasing demand for running complex applications on the edge. These complex applications can be represented as workflows where task dependencies are explicitly specified. To achieve better Quality of Service (QoS), for instance, faster response time and lower energy consumption, computation offloading is widely used in the MEC environment. However, many existing computation offloading strategies only focus on independent computation tasks but overlook the task dependencies. Meanwhile, most of these strategies are based on search algorithms such as particle swarm optimization (PSO), genetic algorithm (GA) which are often time-consuming and hence not suitable for many delay-sensitive complex applications in MEC. Therefore, a highly efficient graph-based strategy was proposed in our recent work but it can only deal with simple workflow applications with linear (namely sequential) structure. For solving these problems, a novel graph-based strategy is proposed for workflow applications in MEC. Specifically, this strategy can deal with complex workflow applications with nonlinear (viz. parallel, selective and iterative) structures. Meanwhile, the offloading decision plan with the lowest energy consumption of the end-device under the deadline constraint can be found by using the graph-based partition technique. We have comprehensively evaluated our strategy using both a real-world case study on a MEC based UAV (Unmanned Aerial Vehicle) delivery system and extensive simulation experiments on the FogWorkflowSim platform for MEC based workflow applications. The evaluation results successfully demonstrate the effectiveness of our proposed strategy and its overall better performance than other representative strategies.